Question

In flow through pipes, according to Darcy-Weisbach formula the loss of head due to friction is proportional to , Where, V = Mean velocity of flow.

• A. $\sqrt{V}$
• B. $V$
• C. $V^2$
• D. $V^3$

Hint:

To remember the Darcy-Weisbach relation, just look at the numerator: $fLV^2$. The V is squared! This is why friction is so much more impactful in high-speed fluid systems.

Answer 1

The Correct Option is C

1. The Darcy-Weisbach Formula: The head loss due to friction ($h_f$) is expressed by the following relationship: $$h_f = \frac{f \cdot L \cdot V^2}{2 \cdot g \cdot d}$$ Where:

• $h_f$: Head loss due to friction (m)

• $f$: Friction factor (dimensionless)

• $L$: Length of the pipe (m)

• $V$: Mean velocity of flow (m/s)

• $g$: Acceleration due to gravity (9.81 m/s$^2$)

• $d$: Diameter of the pipe (m)

2. Relationship with Velocity: From the formula, it is clear that the frictional head loss ($h_f$) is directly proportional to the square of the mean velocity ($V^2$). This means that if the velocity of the fluid in a pipe is doubled, the head loss due to friction will increase by a factor of four ($2^2 = 4$).

3. Practical Significance: This square relationship is critical for engineers when sizing pumps and pipes. High velocities lead to significantly higher energy losses, which is why industrial piping is often designed to keep flow velocities within specific economical ranges to balance pipe cost against energy consumption.